All Questions
21 questions with no upvoted or accepted answers
18
votes
0
answers
734
views
How boundedly generated is $SL_3(\mathbb{Z})$?
The group $G = \mathrm{SL}_3(\mathbb{Z})$ is known to be boundedly generated, that is, there exists some $m \in \mathbb{N}$, and $g_1, \dots, g_m \in G$ such that we have the following equality of ...
- 11.3k
11
votes
0
answers
345
views
Status of questions in "Group Actions on $\mathbb{R}$-trees"?
Culler and Morgan's "Group Actions on $\mathbb{R}$-trees" lists four questions at the end of the introduction. A few have been famously resolved by work of Rips, Bestvina–Feighn and others.
I'm ...
- 1,996
11
votes
0
answers
379
views
Amalgamated product of automatic groups
In Gersten's "Problems on Automatic Groups", Problem 14, he asks the following question: Let $G=A\ast_{C}B$ where $A$ and $B$ are automatic and $C$ is infinite cyclic. Is $G$ automatic?
Is this ...
- 525
7
votes
0
answers
252
views
A conjecture of Lubotzky on ranks of subgroups of special linear groups over the integers
In a 1985 paper named "Dimension function for discrete groups" Lubotzky conjectured that:
For any integer $n \geq 3$ the group $\mathrm{SL}_n(\mathbb{Z})$
contains infinitely many finite index ...
- 11.3k
5
votes
0
answers
200
views
Virtual fibring of $\mathrm{Out}(F_2\times F_2)$
A finitely generated group $G$ is said to virtually fibre if there is a finite index subgroup $H\leq G$ and a non-trivial map $\varphi:H\to\mathbb{Z}$ with $\ker(\varphi)$ finitely generated.
I want ...
- 911
5
votes
0
answers
228
views
Automorphism groups of cocompact Fuchsian groups as mapping class groups
Let $\Gamma$ be a cocompact Fuchsian group. So it has presentation $$\langle x_1,y_1, \dots, x_g,y_g,z_1, \ldots, z_r \mid [x_1,y_1] \cdots [x_g,y_g]z_1 \cdots z_r=1, \ z_i^{m_i}=1 \rangle$$
for some $...
- 8,401
5
votes
0
answers
169
views
In the literature on infinite graphs, are there results on "periodizable" graphs?
Let $G=(V,E)$ be a connected countably infinite $k$-regular simple graph (no loops or multiple edges). For $A$ a finite subset of $V$, let me denote by $G_A=(A,E_A)$ the induced subgraph with vertex ...
4
votes
0
answers
177
views
Ping pong with parabolic isometries on Gromov hyperbolic spaces
For a group $G$ with a non-elementary general type action by isometries on a Gromov hyperbolic geodesic space $(X,d)$, it is well known that you can construct free subgroups of $G$ via the ping pong ...
- 41
4
votes
0
answers
124
views
Abelian-by-cyclic subgroups of exponential growth solvable groups
I am currently looking for a reference to a proof (or counterexample) to the following statement:
Statement: Assume $G$ is a finitely generated solvable group of exponential growth, then there is a ...
- 4,432
3
votes
0
answers
393
views
What about a Cayley n-complex for n>2?
Let $G$ be a finitely presented group. The Cayley graph of the finite generating set is a $1$-complex where the $0$-cells are the elements of $G$ and the $1$-cells are given by the generators (...
3
votes
0
answers
136
views
Existence of loxodromic elements in certain subsets of $\text{PSL}_2(\mathbb C)$
Let $R$ be a subset of $\text{PSL}_2(\mathbb C)$ and consider its natural action on $\mathbb {CP}^1$. We say that $R$ is elementary if either $R$ is conjugated to a subset of $\text{SU(2)}$ or if ...
- 1,058
3
votes
0
answers
493
views
Short exact sequences for amalgamated free products and HNN Extensions
I asked this question on math stackexchange (see here) but didn't get any answer so I thought I would post it here too:
If $A$ and $B$ are groups we have the following short exact sequence:
$$ 0 \to [...
- 721
3
votes
0
answers
209
views
Growth of the number of generators in hyperbolic groups
Let $G$ be an infinite hyperbolic group, and let us further assume that it is residually finite (or even LERF/GFERF) so that we have plenty of subgroups of finite index.
I would like to know if one ...
- 11.3k
3
votes
0
answers
156
views
Cancellations in products of two elements of a hyperbolic group
Let $G$ be a non-abelian free group with the standard generating set and the corresponding word metric. If we take two elements $g,h\in G$ and compute their product $gh$, some letters might cancel, ...
- 637
2
votes
0
answers
68
views
Amplification argument for hyperlinear groups
Let us define a group $G$ to be a hyperlinear group if it satisfies the conclusion of Theorem 3.6. in these notes by Vladimir Pestov. It is well-known that one can use the so-called amplification ...
- 6,224
2
votes
0
answers
110
views
Moment of the hitting measure of a subgroup
Given a [finitely generated] group $G$ and a finite generating set $S$, a measure $\mu$ will have finite $\alpha$-moment if $\sum_{g \in G} \mu(g) |g|_S^\alpha$ is finite (where $|g|_S$ is the word ...
- 4,432
2
votes
0
answers
66
views
Quasi-isometry of solvable minimax groups
[Edits in brackets]
Consider two finitely generated solvable minimax groups $G_i$ ($i = 1,2$) so that $1 \to N_i \to G_i \to Z_i \to 1$ [splits]
with $N_i$ nilpotent, $Z_i$ infinite cyclic and $G_i$ ...
- 4,432
1
vote
0
answers
89
views
The base group of a wreath product of an abelian group by $ {\mathbb{Z}}$ is a characterstic subgroup
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts here can direct me to some relevant results.
Let $A$ be a finitely generated abelian group,...
- 823
1
vote
0
answers
188
views
Does every amenable group $G$ admit a two-sided Folner sequence?
By two-sided Følner sequence I mean a sequence $(F_N)_N$ of subsets of $G$ which is both a left-Følner and a right-Følner sequence.
Context: I just came up with this question and surprisingly I haven'...
- 10.6k
1
vote
0
answers
223
views
Reference on calculation of 2nd cohomology group
Let $G$ be a finitely generated, infinite, countable discrete nonamenable group with zero first Betti number, I.e., $H^1(G, \ell^2(G))=0$, e.g., $G=F_2\times F_2$, the product of free groups of two ...
- 1,528
0
votes
0
answers
105
views
specific qi on free groups
Let $F_n$ be the free group on $n$ generators, $n>1$.
If $\phi$ is a quasi-isometry (or a bijective bilipschitz equivalence) on $F_n$, then what can we say about the explicit form of $\phi$?
In ...
- 1,528